verified 6 stabilizer terms reproduce |qubit_sqrtT>^{ot 5}, checked symbolically
Implied per-copy exponent γ ≤ 0.5170 against a published 0.5000, no improvement.
1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
The six product states φ^5 over the single-qubit stabilizer states are independent in the six-dimensional symmetric subspace (in the Hamming-weight basis they are e0, e5, and the Vandermonde rows (z^w)_w for z = 1, -1, i, -i), so they span Sym^5 and every symmetric five-qubit state, |sqrtT⟩⊗5 among them, has χ ≤ 6; this is the generic value chi_5 = 6 of Lovitz and Steffan (arXiv:2110.07781, Section 1.3). With qubit_sqrtT-m5-lower-6.json (exact census through the Galois-forced subspace) the cell is exact, χ(|sqrtT⟩⊗5) = 6, γ log_2(6)/5 = 0.5170 against the published 1/2. The construction stops here: at six qubits the six product states span only the hyperplane f(1) = f(5) of Sym^6, and every symmetric six-qubit stabilizer state has equal amplitudes on weights 1 and 5, so Sym^6 is not spanned by symmetric stabilizer states and a seven-term decomposition of |sqrtT⟩⊗6, which would beat 1/2, needs non-symmetric terms.
Vincent Russo · this repository (stabrank-research, round 33) · by hand: the six product stabilizer states |0⟩⊗5, |1⟩⊗5, |+⟩⊗5, |→^5, |+i⟩⊗5, |-i⟩⊗5 span Sym^5(C^2); coefficients solved exactly in Q(zeta_16)
{
"schema_version": "0.1",
"orbit": "qubit_sqrtT",
"m": 5,
"direction": "upper",
"rank": 6,
"witness": {
"terms": [
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"l": [
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],
"coeffs": [
"sqrt(2)/8 - sqrt(2)*I/8",
"-sqrt(2)*exp(I*pi/8)/8 + sqrt(2)*exp(5*I*pi/8)/8",
"exp(I*pi/8)/4 + exp(I*pi/4)/4 + exp(3*I*pi/8)/4 + I/4",
"-exp(3*I*pi/8)/4 - exp(I*pi/8)/4 + exp(I*pi/4)/4 + I/4",
"-exp(5*I*pi/8)/4 - exp(I*pi/4)/4 + exp(7*I*pi/8)/4 + I/4",
"-exp(I*pi/4)/4 - exp(7*I*pi/8)/4 + exp(5*I*pi/8)/4 + I/4"
]
},
"provenance": {
"author": "Vincent Russo",
"reference": "this repository (stabrank-research, round 33)",
"method": "by hand: the six product stabilizer states |0>^5, |1>^5, |+>^5, |->^5, |+i>^5, |-i>^5 span Sym^5(C^2); coefficients solved exactly in Q(zeta_16)",
"date": "2026-10-23",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.001,
"wall_clock_hours": 0.001,
"runs": 1,
"hardware": "Apple silicon laptop, one core; direct construction, no search"
}
},
"notes": "The six product states phi^5 over the single-qubit stabilizer states are independent in the six-dimensional symmetric subspace (in the Hamming-weight basis they are e_0, e_5, and the Vandermonde rows (z^w)_w for z = 1, -1, i, -i), so they span Sym^5 and every symmetric five-qubit state, |sqrtT>^5 among them, has chi <= 6; this is the generic value chi_5 = 6 of Lovitz and Steffan (arXiv:2110.07781, Section 1.3). With qubit_sqrtT-m5-lower-6.json (exact census through the Galois-forced subspace) the cell is exact, chi(|sqrtT>^5) = 6, gamma log_2(6)/5 = 0.5170 against the published 1/2. The construction stops here: at six qubits the six product states span only the hyperplane f(1) = f(5) of Sym^6, and every symmetric six-qubit stabilizer state has equal amplitudes on weights 1 and 5, so Sym^6 is not spanned by symmetric stabilizer states and a seven-term decomposition of |sqrtT>^6, which would beat 1/2, needs non-symmetric terms."
}